The tables below show the values of *mPQ* for several values of *x* close to 1. The closer *Q* is to *P*, the closer *x* is to 1 and, it appears from the tables, the closer *mPQ* is to ... . This suggests that the slope of the tangent line ℓ should be *m* =...

fill the plank

To fill in the blanks correctly based on the provided solution and the values in the table, here's how you should complete the sentences:

1. The closer \( Q \) is to \( P \), the closer \( x \) is to 1 and, it appears from the tables, the closer \( m_{PQ} \) is to \( 16 \).
2. This suggests that the slope of the tangent line \( \ell \) should be \( m = 16 \).

Would you like any additional details or have any questions?

### Suggested Questions:
1. How do you derive the equation of a tangent line using limits?
2. Can you explain the concept of a secant line?
3. How do you use the point-slope form to write the equation of a line?
4. What does it mean for a function to have a derivative at a point?
5. How does the limit process help in finding the slope of the tangent line?

### Tip:
When finding the equation of a tangent line, remember that the slope of the tangent line at a point is the derivative of the function evaluated at that point.

Explore how tangent lines relate to limits in calculus with a detailed explanation. Learn how to determine the slope of a tangent line using the derivative as a limit definition. Suitable for advanced high school students and those interested in calculus concepts.

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